A continuously varying family of one-dimensional vector spaces over a topological space. A related tag is the vector-bundles tag.

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Are two line bundles with the same ramification type necessarily isomorphic?

I have no motivation for the following problem, I am just curious if it is true or not. Here it is: If $l_1$ and $l_2$ are two complete $g^r_d$'s on a smooth curve $C$ such that the vanishing ...
2
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2answers
228 views

Could we extend any line bundle on the smooth part of a singular curve to a line bundle on the whole curve?

Let $X$ be a singular curve over an algebraic closed field $k$ with characteristic zero. Let $Z$ be the closed subset of singular points on $X$ and $U=X-Z$ be the smooth part, which is an open subset ...
6
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2answers
306 views

Will (general points + small number of arbitrary points) impose independent condtions on plane curves?

It is well known that imposing vanishing at general points of $\mathbb P^2$ gives independent conditions on curves of degree $d$. Also, it is known that a small number ($\le d+1$) points always impose ...
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324 views

Correspondence between line bundles and $U(1)$-bundles: a possible mistake from physicists? [closed]

I am reading a paper written by physicists and they say the following: Let $(L,h,\nabla)$ be an holomorphic Line bundle equipped with a hermitian metric $h$ and Chern connection $\nabla$. If ...
2
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1answer
163 views

Computing Euler Characteristics of Line Bundles on the Hilbert Scheme of n points

Let $S^{[n]}$ be the Hilbert scheme of $n$ points on a smooth projective surface (actually, right now I am particularly interested in del Pezzo surfaces). Let $B$ be the exceptional divisor of the ...
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1answer
325 views

Is there a toplogically trivial line bundle over a compact Riemann surfaces that isn't holomorphically trivial? [closed]

Is a complex line bundle over a compact Riemann surface topologically trivial iff it is holomorphically trivial? If so, how does one demonstrate that, and if not, what is a counterexample?
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3answers
439 views

Movable Divisors

Let $X$ be a projective variety. Does anyone know an example of a movable reducible divisor $D\in Mov(X)$ such that any element in the linear system $|D|$ of $D$ is reducible?
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4answers
738 views

Cone over the Veronese surface

Let $V\subset\mathbb{P}^5$ be the Veronese surface and let $X\subset\mathbb{P}^6$ be the cone over it. Since $X$ is $\mathbb{Q}$-factorial there are two integers $a,b$ such that $aK_X = ...
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114 views

How to see that this pairing of line bundles is multiplicative?

Given a projective flat morphism $p: X \rightarrow Y$ of integral noetherian schemes of relative dimension one. For a coherent sheaf $F$ on $Y$ we can define a line bundle $det(F)$ on $Y$ and for a ...
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110 views

When can one find holomorphic sections vanishing at a point to a certain order?

Let $X$ be a compact complex manifold (say of dimension $2$) and $L \rightarrow X $ a holomorphic line bundle. Consider the following statements: Statement $A_0$: Given any point $p\in X$, there ...
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0answers
102 views

Computing Euler Charactistics of Line bundles on Hilbert Schemes of points on Surfaces

Let $S^{[2]}$ be the Hilbert scheme of two points on a smooth projective surface (actually, right now I am particularly interested in del Pezzo surfaces). Let $B$ be the exceptional divisor of the ...
2
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2answers
165 views

Holomorphic Line Bundles over a Homogeneous Space

Let $M=G/H$ be (compact) homogeneous complex manifold, and let $L$ be a line bundle over $M$. Can one always equip $L$ with a holomorphic structure? Can there be more then one such holomorphic ...
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1answer
121 views

Perfectness of the Jacobian of a curve

Let $C$ be a smooth projective curve over a field $K$ of characteristic $0$ (but not necessarily algebraically closed). Let $\mathcal{L}$ be a line bundle on $C$ of degree $0$. Fix an integer ...
15
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1answer
474 views

Maps to projective space == line bundles; what do maps to weighted projective space correspond to?

A map from an algebraic variety $X$ to a projective space is the same thing as a globally generated line bundle on $X$. What geometric object on $X$ corresponds to a map to a weighted projective ...
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1answer
128 views

Big divisors and small transformations

Let $X$ be a smooth projective variety such that $-K_X$ is ample. Let $f:X\dashrightarrow Y$ be a small $\mathbb{Q}$-factorial transformation. I would like to know if is true or not that: $-K_Y$ is ...
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2answers
603 views

Big and Nef divisors

In Example 2.2.19 of Lazarsfeld, Positivity in Algebraic Geometry I, I found the following statement: Let $D$ be a divisor on an irreducible projective variety $X$. Then $D$ is nef and big if and ...
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1answer
79 views

Two questions about line bundles over Kuranishi families

i'm studying the article "Variétés Kahleriennes dont la première classe de chern est nulle" by Arnaud Beauville and i have a couple of questions i would like to ask you, hoping they are not too ...
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1answer
107 views

When is the Clifford index of a curve computed by pencils?

Under which circumstances is the Clifford index of a curve computed by pencils?
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2answers
185 views

Restriction of a line bundle to a two-cycle

I am reading a paper on Chiral Differential Operators http://arxiv.org/pdf/hep-th/0604179v3.pdf and it says on page 23 that a line bundle over a manifold C can be characterized completely by its ...
0
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1answer
196 views

Birkhoff decomposition vanishing of the Chern numbers

Birkhoff decomposition vanishing of the Chern numbers of the holomorphic line bundles of the Birkhoff-Grothendieck decomposition, is some statement I read off in One of Connes papers. Without going ...
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1answer
97 views

Unicity of a vector field on $S^1$-bundle

Let M be a complex smooth manifold,and let $\zeta $ be a vector filed on $M$, why always there exists a unique vector field $\hat{\zeta }$ on $L^{\times}$ which project down to $\zeta $ and $\alpha( ...
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4answers
324 views

An isomorphism on space of smooth sections

Let $M$ be a smooth complex manifold and $L$ be a complex line bundle over $M$. Let $\Gamma(M,L)$ be the space of smooth sections. Why $\Gamma(M,L)$ is it isomorphic to $$A=\{f:L^{\times}\to ...
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2answers
326 views

Isomorphism of connections on a complex line bundle

Reading an article I faced with the following theorem, please give me a reference to a proof of the fact which is stated without any reference in the article. Is it a well-known fact? Theorem. Let $E ...
0
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2answers
575 views

Uniqueness on square root of complex Line Bundle

Let $L$ be a line bundle over a compex manifold $X$, a square-root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. My question is when the square-root of Line Bundle is unique?
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142 views

Hopf lemma for line bundles on curves in algebraic geometry

In the paper http://arxiv.org/pdf/math/0110256v1.pdf Claire Voisin proves that all linear subspaces which lie inside of a (not too big) secant variety of a smooth projective curve must lie inside one ...
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125 views

canonical bundle of the relative spectrum

maybe it is a very trivial quetion but: suppose we have a smooth projective variety $X$ over $k$ and $\mathcal{A}$ an $\mathcal{O}_X$ algebra. We have the relative spectrum ...
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1answer
216 views

Could we construct the Jacobian variety of a smooth curve $C$ with genus $>2$ from its derived category $D(C)$?

Let's consider a smooth curve $C$ over $\mathbb{C}$. We know that the Jacobian variety $Jac(C)$ of $C$ is the moduli space of the degree $0$ line bundles on $C$. $Jac(C)$ is an abelian variety of ...
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308 views

Ample Line Bundles on Algebraic Spaces

The sources known to me (Knutson's Algebraic Spaces and Pascual-Gainza's Ampleness criteria for algebraic spaces) define a line bundle $L$ on an algebraic space $X$ (over a base scheme $S$) to be ...
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1answer
334 views

A reference about Dolbeault cohomology

I am looking for a reference about Dolbeault cohomology when the line bundle is not supposed to be positive.
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1answer
228 views

extension of torsion line bundles

Assume that you are given a smooth projective algebraic variety $X$ and a divisor $D$ with normal crossings on it. Put $U=X-D$. Assume now that $L$ is a torsion line bundle(=invertible sheaf) on $U$, ...
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1answer
463 views

Vanishing theorems in positive characteristic

In the paper Deligne, Pierre; Illusie, Luc (1987), "Relèvements modulo $p^{2}$ et décomposition du complexe de De Rham", Inventiones Mathematicae 89 (2): 247–270, doi:10.1007/BF01389078 I found the ...
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3answers
2k views

What is Kirillov's method good for?

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...
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1answer
147 views

Trivial Line Bundle-Riemann surfaces

What are the Hermitian metrics in a trivial line bundle on a Riemann surface X? I read that a Hermitian metric in the trivial line bundle is equivalent to a $\mathcal{C}^{\infty}$ weight function ...
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0answers
201 views

canonical model of a reducible curve

Let $C$ be a stable reducible curve. Is there a natural way to define it's canonical model (I guess via the dualizing sheaf)? And does somehow the dualizing sheaf restrict to the (probably twisted) ...
3
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2answers
339 views

Line bundles on K3 surfaces

Let $L$ be a line bundle on an (algebraic) K3 surface over a field $k$. The Riemann-Roch theorem specializes to $$ \chi(X, L)=\frac{1}{2}(L\cdot L)+2 $$ which can be rewritten as $$ h^0(X, ...
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2answers
773 views

How many flat connections has a line bundle in algebraic geometry?

Suppose $X$ is a projective variety over $\mathbb C$. I am happy to entertain more or different adjectives — I'm not looking for the most general statement, but rather to understand when and how ...
2
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1answer
421 views

On morphisms to projective space arising from a linear system

Context: This question arose as I was reading the proof of Application 6.1 in Mumford's Abelian Varieties. However, I have extracted all of the relevant information below so this question should ...
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1answer
230 views

Extending line bundles

Suppose you have a one parameter family of algebraic varieties over unit disk, such that the central fiber is singular and is a union (normal-crossing) of two varieties and the rest are smooth. Is it ...
2
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1answer
190 views

divisors and powers of line bundles

Can anyone help me with the following question? Let $X$ be a smooth, projective algebraic variety over a field $k$ of characteristic zero. Let $D$ be an effective divisor on $X$ and $m \geq 2$ an ...
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0answers
203 views

How does the line bundles look like on a proper model (or Néron model) of an abelian variety?

How does the line bundles look like on a proper model (or Néron model) of an abelian variety? Who knows references about this? In particular, let us work over a trait $S=\mathrm{Spec} R$, where $R$ ...
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1answer
523 views

Non-compact Kähler manifolds which admit a positive line bundle

A complex manifold which admits a positive line bundle is automatically Kähler. Furthermore, if the manifold is compact, then it is projective by the Kodaira Embedding Theorem. In particular, not ...
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108 views

pairing theta functions for different complex structures

I apologize for my previous attempt to ask this, which was very badly written. Let us start with $\mathbb{C}\times\mathbb{C}$. To form an Hermitian line bundle over a complex torus with complex ...
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0answers
296 views

Theta functions and Fourier transforms

Let $T_\tau$ be the 2-dimensional torus, with the complex structure induced by the lattice generated by $1$ and $\tau$. Then for a line bundle $L_k$ over $T$ with level $k$, there is an orthonormal ...
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votes
4answers
1k views

Cohomology of line bundles

For sure answers to my questions are well known - but I never saw them anywhere. Let $X$ be a smooth projective (or just proper) variety over an algebraically closed field $k$. Let $A_i$ be the ...
7
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3answers
408 views

line bundles and universal covers

When dealing with some lifting problems, I came across the following problem, which probably has a well-known answer, but anyway: Suppose I have a (locally contractible) topological group $G$, such ...
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3answers
351 views

Pedagogical notes on line bundles on complex projective manifolds

I would like to find some notes (or book), that explains on a very basic level what is a line bundle on a complex projective manifold. Maybe even, what is a line bundle on $\mathbb CP^n$. It seems ...
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1answer
631 views

Is there a mathematical explanation for the Aharonov-Casher effect?

Recall that the Aharonov-Bohm effect can be interpreted mathematically as follows. Consider an electromagnetic field A on some smooth manifold M, i.e., A is an element in the first differential ...
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1answer
210 views

etale covers of line bundles on an abelian variety

subj: etale covers of line bundles on an abelian variety Is there an explicit decryption of finite etale covers of a line bundle $L$ on an abelian variety and its associated C*-bundles $L^o = L ...
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1answer
779 views

Connections on line bundles over the torus

If I understand correctly, every line bundle $L$ over the (2-dim) torus can be obtained from a quotient of $\mathbb{R}^2 \times \mathbb{C}$ by a $\mathbb{Z}^2$ lattice action. Different line bundles ...
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2answers
611 views

What is the geometric point of view of an algebraic line bundle compared to a analytic line bundle?

Hi folkz, I'm trying to learn more about line bundles, invertible sheaves and divisors on schemes. I understand the connection beweteen Cartier and Weil Divisors and the connection between Cartier ...